On a problem in simultaneous Diophantine approximation: Schmidt's\n conjecture

2010/01/15 by Dzmitry Badziahin, Badziahin, Dzmitry, Andrew Pollington +3
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Advanced Mathematical Theories and Applications #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1001.2694

Abstract

For any i,j \≥ 0 with i+j =1, let bad(i,j) denote the set of points\n(x,y) \∈ R² for which \max \‖qx\‖^1/i, \‖qy\‖^1/j > c/q for\nall q \∈ N . Here c = c(x,y) is a positive constant. Our main result\nimplies that any finite intersection of such sets has full dimension. This\nsettles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous\nDiophantine approximation.\n

Citations

• Fractal Geometry: Mathematical Foundations and Applications.

Related

• Intersective polynomials and Diophantine approximation, II
• On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem
• Positive integers: counterexample to W.M. Schmidt's conjecture
• Diophantine approximations with positive integers: some remarks
• Multiplicative Diophantine approximation with restricted denominators